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Chapter 5 Study Guide

by: idk

Chapter 5 Study Guide Math M-118

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Covers all of Chapter 5 materials and examples!
Finite Math
George O'Donnell
Study Guide
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This 7 page Study Guide was uploaded by idk on Monday November 2, 2015. The Study Guide belongs to Math M-118 at Indiana University taught by George O'Donnell in Summer 2015. Since its upload, it has received 67 views. For similar materials see Finite Math in Mathematics (M) at Indiana University.

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Date Created: 11/02/15
5.2:  Expected  Value  and  Standard  Devia▯on   Ø  The  main  use  of  sta▯s▯cs  is  to  collect  and  analyze  data.     Ø  Every  data  set  of  an  experiment  contains  two  important  pieces  of   informa▯on  which  can  be  used  to  describe  the  results  of  the   experiment.    These  two  descrip▯ve  sta▯s▯cs  are:      (1)  Expected  Value    (denoted  by:    E[x]    or    μ)      -­‐tells  us  the  average  value  of  the  experiment        (2)      Standard  Devia▯on    (denoted  by:    σ)      -­‐tells  us  how  much  the  data  varies  from  the  average  value       Ø  In  order  to  calculate  these  two  pieces  of  informa▯on,  we  need  to  know   what  the  experiment  is  trying  to  measure  (i.e.  the  variable  of  the   experiment).   The  Random  Variable  of  an  Experiment   Ø   For  every  possible  outcome  of  an  experiment,  a  number  can  be  assigned.    This     number  represents  whatever  we  are  measuring  or  trying  to  keep  track  of  in  the     experiment.         Ø   The  numbers  that  are  assigned  to  each  outcome  of  an  experiment  are              referred  to  as  the  values  of  the  random  variable    (  X  )   Example:   if  we  roll  a  die  three  ▯mes,  we  could  keep  track  of  how  many  ▯mes     a  ‘5’  appears.    So,  our  random  variable,  X,  assigns  a  number  (the  number  of   5’s  that  occur  in  three  rolls)  to  each  outcome  in  the  sample  space.       Therefore,      =  {0,  1,  2,  3},  since  the  possibili▯es  range  from  obtaining  no  ‘5’s   to  the  possibility  of  obtaining  a  ‘5’  on  all  three  rolls.   Types  of  Random  Variables     Ø     In  Sec▯on  5.2,  we  will  focus  on  experiments  that  have  discrete  random     variables  (an  experiment  with  a  countable  or  finite  number  of  possible     outcomes)        A  special  type  of  discrete  random  variable  is  called  a  binomial  random     variable.    Any  experiment  which  is  a  sequence  of  Bernoulli  trials  (4.4)  is  said     to  contain  a  binomial  random  variable.   Ø So,  the  experiments  in  Sec▯on  5.2  will  contain  either:   1)  a  binomial  random  variable  (if  the  experiment  is  a  sequence  of                  Bernoulli  trials)        OR   2)    a  non-­‐binomial  random  variable  (if  the  experiment  is  not  a                sequence  of  Bernoulli  trials)   Ø   Remember  from  Sec▯on  4.4,  for  an  experiment  to  be  a  sequence  of  Bernoulli     trials,  it  must  sa▯sfy  three  condi▯ons:   1)  Each  trial  can  result  in  only  two  possible  outcomes  –  success  or  failure   2)  The  trials  of  the  sequence  are  independent  of  each  other   3)  For  each  trial,  the  probability  of  success  is  always  the  same   If  an  experiment  fails  to  meet  all  four  condi▯ons,  then  the  variable  of  the     experiment  is  not  a  binomial  random  variable.   Random  Variables  -­‐-­‐  Examples   For  each  experiment,  state  whether  the  random  variable  is  binomial  or  not  binomial.     Also,  list  all  possible  values  of  the  random  variable  X:   (a)   There  are  6  marbles  in  a  bag.    4  are  blue  and  2  are  red.    Three  marbles  are     selected  at  random.    A  random  variable  X  is  defined  as  the  number  of  red  selected.   Not  Binomial  since  the  experiment  does  not  consist  of  independent  trials.   X  =  {0,1,2}   (b)   A  fair  pair  of  dice  is  rolled  5  ▯mes.    Let  the  random  variable  X  be  defined  as  the     number  of  ▯mes  the  dice  sum  to  7.   Binomial  since  the  experiment  consists  of  mul▯ple  independent  trials.   X  =  {0,1,2,3,4,5}   (c)   An  experiment  consists  of  randomly  selec▯ng  two  coins  from  a  box  containing   2  half  dollars  and  2  quarters.    A  random  variable  X  is  defined  as  the  value  of  the     coins  in  cents.   Not  Binomial  since  the  experiment  does  not  consist  of  independent  trials.     Also,  a  success/failure  is  not  defined.    X  =  {100,75,50}   (d)  A  fair  coin  is  flipped  10  ▯mes.    Let  the  random  variable  X  be  the  number  of  heads   that  appear.   Binomial  since  the  experiment  consists  of  mul▯ple  independent  trials.   X  =  {0,1,2,3,4,5,6,7,8,9,10}   Expected  Value  (for  binomial)          For  experiments  that  contain  a  binomial   random  variable,  we  have  a  very  short   formula  that  computes  the  expected  value:                             n·p                              E[x]  =   Where: n =  #  of  trials  of  the  experiment   p =  probability  of  obtaining  a  success  per          trial     Examples:    Find  E[x]  (for  a  Binomial  Random  Variable)   a)  An  experiment  consists  of  flipping  a  fair  coin  20  ▯mes  and  no▯ng  the       number  of  tails  that  occur.  Find  the  expected  value  of  the  experiment.   E[x] = n·∙p     n    =     20   p    =     = (20)·∙(1/2)     = 10 tails   b)  An  experiment  consists  of  flipping  an  unfair  coin  20  ▯mes  (with                      Pr[Tails]  =  ¼)  and  no▯ng  the  number  of  tails  that  occur.    Find  the   expected  value  of  the  experiment.   E[x] = n·∙p     n    =     20   p    =     = (20)·∙(1/4)     = 5 tails   c)  An  experiment  consists  of  rolling  a  fair  pair  of  dice  12  ▯mes,  and                    no▯ng  the  number  of  ▯mes  the  dice  match.    Find  the  expected  value   of  the  experiment.   E[x] = n·∙p     n    =     12   p    =     = 1/6   = (12)·∙(1/6)     = 2 matches   Expected  Value  for  a  Non-­‐Binomial   Ø  To  find  the  expected  value  for  an  experiment  with  a  non-­‐binomial  random   variable  (that  is,  an  experiment  that  is  not  Bernoulli)  we  cannot  use  the   formula.       Ø  Rather,  to  obtain  the  expected  value  we  have  to  create  a  table.   Ø  The  table  is  called  a  Probability  Density  Func▯on     Crea▯ng  a  Probability  Density  Func▯on:   Ø  In  the  first  column  of  the  table,  we  list  all  of  the  possible  values  of  x  (the   random  variable)     Ø  In  the  next  column  we  list  the  probabili▯es  for  each  value  of  x (denoted  by   Pr[x]).     Ø  Let’s  see  how  this  is  done.  


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